While calculating the total number of chess games possible in the universe I used the following approach:
$ \binom{64}{32} $ to select the total number of ways I can arrange 32 pieces on the 64 squared chessboard
$ \binom{32}{16} $ to select 16 places where I can place a specific player's pieces on the board
$ \frac{16!}{8!\cdot(2!)^{3}} $ this gives the ways we can arrange a specific color's pieces in these selected 16 squares. This will also be true for the opponent's peices therefore this result is to be multiplied two times
Hence my final result becomes:
$ \binom{64}{32} \cdot \binom{32}{16} \cdot \frac{16!}{(8!)(2!)^{3}} \cdot\frac{16!}{(8!)(2!)^{3}} $
But this would contain a lot of illegal positions like both kings in check, pawns on the first ranks etc. So how do I remove these posisitons from the extremely large number I have calculated